David Moyes said his team only had themselves to blame. Photograph: Darren Staples/Reuters

The Everton manager, David Moyes, admitted his team only had themselves to blame after losing to their Merseyside rivals Liverpool in the FA Cup semi-final at Wembley.

"It was our own fault today because we gave them the opportunities," a downcast Moyes told ESPN.

"I thought we were always going to have to deal with Liverpool in the second half. They were 1-0 down in the semi-final of the Cup – if it was the other way round we'd be the same. Liverpool started brightly. I thought we'd weathered it and hoped we might get a second. As it was we just got ourselves into a wee bit of trouble."

Everton, however, by no means had a monopoly on mistakes; the veteran Liverpool defender Jamie Carragher's slip had allowed Nikica Jelavic to put Moyes's men into a first-half lead.

Asked to comment on the Reds' victory, Carragher told ESPN: "Relief really. Liverpool losing to Everton in the semi-final – it would have been a nightmare."

"Everyone wanted Everton to win. David Moyes said the whole country was with them. Well, we showed we're a special club, special fans. Not many teams can come back. We've done it again. We've got that ability to come back."

Carragher and the goalkeeper Brad Jones were both quick to praise the strike force, particularly Andy Carroll after his second crucial late winning goal this week.

"He'll be remembered forever for that," the former England international said. "And [Luis] Suárez I wouldn't swap for any other player in the league."

Carroll, who also scored Liverpool's winner in a midweek Premier League match against Blackburn Rovers, had struggled to live up to his £35m transfer from Newcastle United last year.

"It's the best feeling ever," Carroll said. "We worked hard and getting the goal right there at the end was great."

The match between the Liverpool rivals was played a day before the 23rd anniversary of the Hillsborough disaster in which 96 Liverpool fans died. It was the first Wembley encounter between the teams since the 1989 FA Cup final, which Liverpool won just weeks after Hillsborough.